标量守恒律的消失粘性最小移动Fourier谱方法

中文摘要 AI 辅助

我们提出了一种用于周期边界条件下标量双曲守恒律的消失粘性最小移动（VVMM）Fourier谱方法。该方法是一种全离散时空残差最小化格式，在空间上采用Fourier谱离散，在时间上采用多项式离散，并通过消失粘性和热核后处理进行稳定。非线性守恒律的Fourier离散化中的一个基本分析难点是Fourier投影与非线性通量之间的相互作用；由此产生的投影残差包含未分辨的高频分量。VVMM通过在每个时间片上选择使惩罚截断粘性守恒律残差最小化的数值解来解决这一问题，然后应用热核后处理来阻尼未分辨模态。我们证明了在任意固定空间维度下的全离散$L^1$误差估计，该估计将消失粘性误差、时间插值误差和空间Fourier截断误差分离。在惩罚截断残差的精确最小化和适当的参数耦合下，该估计对每个固定的$\gamma>0$给出$L^1$收敛速率$N^{-1/2+\gamma}$，其中常数依赖于$\gamma$。一维和二维数值实验展示了稳定的激波捕捉，没有可见的Gibbs振荡，且收敛行为与理论一致。

英文摘要

We propose a vanishing-viscosity minimizing movement (VVMM) Fourier spectral method for scalar hyperbolic conservation laws under periodic boundary conditions. The method is a fully discrete space--time residual-minimizing scheme, Fourier spectral in space and polynomial in time, stabilized by vanishing viscosity and heat-kernel postprocessing. A basic analytical difficulty in Fourier discretizations of nonlinear conservation laws is the interaction between Fourier projection and the nonlinear flux; the resulting projection residual contains unresolved high-frequency components. VVMM addresses this issue by selecting, on each time slab, the numerical solution that minimizes {a penalized cut-off viscous conservation laws residual} and then applies heat-kernel postprocessing to damp unresolved modes. We prove a fully discrete $L^1$ error estimate, in any fixed space dimension, that separates the vanishing-viscosity error, the temporal interpolation error, and the spatial Fourier truncation error. {Under exact minimization of the penalized cut-off residual and suitable parameter coupling,} the estimate yields, for each fixed $γ>0$, an $L^1$ convergence rate $N^{-1/2+γ}$, with a constant depending on $γ$. Numerical experiments in one and two dimensions illustrate stable shock capturing without visible Gibbs oscillations and convergence behavior consistent with the theory.

发表机构

• The Hong Kong Polytechnic University（香港理工大学）

机构由 AI 辅助整理，请以论文原文为准。

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26pages, 16 figures